sevensixtwo 18 Posted October 5 Zero is the same as zero hat. I use the hat to make a formal distinction between R and R_0 so that R_0 mirrors hat R in the notation. I don't see what you're getting at in your posted reasoning, nor do I feel like I understand what you have proven (attempted to prove) with it so I can neither agree nor disagree. 0 Share this post Link to post Share on other sites

uncool 202 Posted October 5 (edited) Studiot: " Finally he comes at the end to state that if he can find a number, off the critical line, whose zeta function is zero then he has disproved the theorem. That is self evident. But he has to show that (at least) one such number exists." He does show such a "number" "exists" by abusing properties of infinity. sevensixtwo: 1) You cannot "separate" the real numbers into a neighborhood of zero and a neighborhood of infinity, so that each real number lies in exactly one of the two. 2) A neighborhood of infinity is not usually defined in the way you have chosen; the way you have chosen is inconsistent with the usual properties of the real numbers. 3) The Riemann hypothesis is about the complex numbers. If you want to talk about it anywhere else (for example, in this "neighborhood of infinity"), then you are dealing with a problem different from the Riemann hypothesis. 4) If you are going to shoo people off to links about papers that have been discussed here already, you probably should address the criticisms of those papers, too. Edited October 5 by uncool 0 Share this post Link to post Share on other sites

taeto 37 Posted October 5 9 hours ago, sevensixtwo said: That doesn't follow. If you think it does, please give two real numbers whose difference is infinity. Before you do that, please consider that if you define a real number as numeral rather than as a cut in the real number line I will have to remind you that the definition of a real number is that it is a cut in the real number line. I just pointed out that \(\widehat{\infty}\) is the difference between \(b\) and the real part of \(z_0\), and deduce that \(\widehat{\infty}\) is also real. Are you sure about your definition of a real number, because it sounds circular? 0 Share this post Link to post Share on other sites

studiot 1471 Posted October 5 (edited) 4 hours ago, sevensixtwo said: Zero is the same as zero hat. I use the hat to make a formal distinction between R and R_0 so that R_0 mirrors hat R in the notation. I don't see what you're getting at in your posted reasoning, nor do I feel like I understand what you have proven (attempted to prove) with it so I can neither agree nor disagree. Thank you for responding to my question. Here is your first sentence. Quote Let R be separated between real numbers in the neighborhood of the origin [math]{\hat 0}[/math] and real numbers in the neighborhood of infinity [math] \pm \hat \infty [/math]. I asserted that this is ambiguous and you say you can't see why. So here it is in detail. Let R be separated between the real numbers in the neighborhood of the origin zerohat and....... So does this mean zerohat refers to the neighborhood or to the origin itself ? That first part of the sentence could be taken to mean either. So it is ambiguous. The difference is vital to the rest of the working. Since you very shortly write equation 1 which is invalid if zerohat refers to the neighborhood I did you the courtesy of assuming it refers to the origin itself. This then leaves the issue of why you have chosen to provide two different but similar symbols for the same thing. Surely a recipe for confusion. 2 hours ago, uncool said: Studiot: " Finally he comes at the end to state that if he can find a number, off the critical line, whose zeta function is zero then he has disproved the theorem. That is self evident. But he has to show that (at least) one such number exists." Thank you for your comment and also the ones about the working. It is difficult to sort out when the working refers to a set (neighborhood) and when a real number I agree with your comment about the separation of the reals into neighborhoods, which is why I showed the boundary set as 's' on my sketch. Had the OP not already selected R_{b} for something else I could have used that. @sevensixtwo As I understand neighborhoods as applied to the reals, the infinity and its neighborhood arises not as an endpoint to the real line but as a result of a limiting process for such expressions as [math]\mathop {\lim }\limits_{x \to \infty } \frac{{\left( {x - 1} \right)}}{{\left( {x + 1} \right)}} = 1[/math] But you should certainly listen to uncool, analysis is much more his area of Mathematics than mine. Edited October 5 by studiot 0 Share this post Link to post Share on other sites

taeto 37 Posted October 5 1 hour ago, studiot said: As I understand neighborhoods as applied to the reals, the infinity and its neighborhood arises not as an endpoint to the real line but as a result of a limiting process In one of the first posts I already asked him to define a neighbourhood of infinity, which he did by the example of the complex plane, where it is given as the unbounded region outside a circle of radius R. The real line is special, since there are two regions defined by a circle \( \{-R,R\}\), a negative and a positive neighbourhood of infinity. In higher dimension the neighbourhood is always a connected set. But I suspect that he is thinking of a neighbourhood of infinity as something different yet. 0 Share this post Link to post Share on other sites

studiot 1471 Posted October 5 45 minutes ago, taeto said: In one of the first posts I already asked him to define a neighbourhood of infinity, which he did by the example of the complex plane, where it is given as the unbounded region outside a circle of radius R. The real line is special, since there are two regions defined by a circle {−R,R} , a negative and a positive neighbourhood of infinity. In higher dimension the neighbourhood is always a connected set. But I suspect that he is thinking of a neighbourhood of infinity as something different yet. Thank you. There are indeed multiple issues later in the 'proof'. But at the stage I am discussing, only the reals and a single dimension have been introduced. Whatever is said about this needs to be self consistent, unambiguous and preferably in line with the simplest conventions in normal use. It is noticable that mathematicians on other full blooded mathematics forums are being confused by the definitions and notation employed in the OP's presentation. 0 Share this post Link to post Share on other sites

sevensixtwo 18 Posted October 5 (edited) 7 hours ago, uncool said: 1) You cannot "separate" the real numbers into a neighborhood of zero and a neighborhood of infinity, so that each real number lies in exactly one of the two. 2) A neighborhood of infinity is not usually defined in the way you have chosen; the way you have chosen is inconsistent with the usual properties of the real numbers. 3) The Riemann hypothesis is about the complex numbers. If you want to talk about it anywhere else (for example, in this "neighborhood of infinity"), then you are dealing with a problem different from the Riemann hypothesis (1) Yes I can. You can make your same argument about "cannot" regarding the square root of a negative number and yet we use those every day. (2) I use exactly the usual definition of a neighborhood of infinity. You are totally wrong. If there is some inconsistency with the definition of a real number number as a cut in the real number line then I challenge you to identify it. If there is some other inconsistency besides that, then those are only properties of real numbers in the neighborhood of the origin. (3) I am discussing a region of the complex plane which has been previously neglected but the neighborhood of infinity is absolutely in the complex plane. Again, you are totally wrong. 6 hours ago, taeto said: I just pointed out that ∞ˆ is the difference between b and the real part of z0 , and deduce that ∞ˆ is also real. Are you sure about your definition of a real number, because it sounds circular? Thank you. I agree that (infinity - b) + b = infinity , but I do not see how that implies that infinity is a real number. My definition of a real number is the one I learned in the first five minutes of the first lecture of my first semester of undergraduate real analysis: a real number is a cut in the real number line, only that and nothing more. All other definitions have to do with fields of numbers and other more complicated things that exceed the definition of a real number. Edited October 5 by sevensixtwo 0 Share this post Link to post Share on other sites

taeto 37 Posted October 5 1 hour ago, sevensixtwo said: but I do not see how that implies that infinity is a real number. My definition of a real number is the one I learned in the first five minutes of the first lecture of my first semester of undergraduate real analysis: a real number is a cut in the real number line, only that and nothing more. All other definitions have to do with fields of numbers and other more complicated things that exceed the definition of a real number. I do not recognize your definition of a real number. But it is not serious, because after all the precise definition does not matter, so long as we agree about the properties of real numbers. So at which step do I go wrong in the following: 1. if x is a real number, then -x is a real number, 2. the difference y-x is the same as y+(-x), 3. if x and y are real numbers, then x+y is a real number? Because if these are all correct, then the difference y-x between real numbers x and y will always be a real number. 0 Share this post Link to post Share on other sites

sevensixtwo 18 Posted October 5 (edited) I agree that the rules you have proposed lead to the implication that infinity is a real number but I am not willing to concede that these rules are part of the definition of a real number. For that, you would need to present some third party source which say the rules you posed are part of the definition of real numbers rather than merely some properties which can be derived by only considering real numbers in the neighborhood of the origin. Until you do, I will stick with the definition in my real analysis book. If you have such a book it is probably in yours too: real numbers are cuts in the real number line. I just tried to google it and Wolfram doesn't define real numbers but I found the same definition on Wikipedia. I can post the link to Wikipedia if you want. Since you seem to have a brain and are not trolling me like user name "stupid idiot," what is your opinion on an implication that infinity is a real number? Edited October 5 by sevensixtwo -3 Share this post Link to post Share on other sites

studiot 1471 Posted October 5 1 hour ago, sevensixtwo said: I agree that the rules you have proposed lead to the implication that infinity is a real number but I am not willing to concede that these rules are part of the definition of a real number. For that, you would need to present some third party source which say the rules you posed are part of the definition of real numbers rather than merely some properties which can be derived by only considering real numbers in the neighborhood of the origin. Until you do, I will stick with the definition in my real analysis book. If you have such a book it is probably in yours too: real numbers are cuts in the real number line. I just tried to google it and Wolfram doesn't define real numbers but I found the same definition on Wikipedia. I can post the link to Wikipedia if you want. Since you seem to have a brain and are not trolling me like user name "stupid idiot," what is your opinion on an implication that infinity is a real number? I don't know why I should tell you this since you have been arrogantly dismissive towards my comments and now fail completely to address them, in defiance of the rules here. But your textbook tells you only part of the story. There were originally two definitions of the real numbers, coming from as opposite approaches as possible (synthetic v analytic) due to Cantor and Dedekind. Your textbook is offering you what are known as Dedekind cuts. They were later shown to be equivalent. A good textbook will do the same. see here. https://www.google.co.uk/search?source=hp&ei=76G3W6GkB-WprgTA-aqoDg&q=equivalence+of+dedekind+and+cantors+real+numbers&oq=equivalence+of+dedekind+and+cantors+real+numbers&gs_l=psy-ab.3...1250.22502.0.22698.50.48.0.2.2.0.276.6196.0j36j2.38.0....0...1c.1.64.psy-ab..10.31.4860...0j0i131k1j0i10k1j0i22i30k1j0i22i10i30k1j33i22i29i30k1j33i160k1j33i10i160k1j33i10k1j33i21k1j33i10i21k1.0.8SA5CoIrGpU 0 Share this post Link to post Share on other sites

Phi for All 5229 Posted October 5 1 hour ago, sevensixtwo said: Since you seem to have a brain and are not trolling me like user name "stupid idiot," what is your opinion on an implication that infinity is a real number? ! Moderator Note You will be civil here or you'll be banned. It's our #1 rule. We attack ideas, not people. Shape up. 0 Share this post Link to post Share on other sites

taeto 37 Posted October 5 (edited) 2 hours ago, sevensixtwo said: I agree that the rules you have proposed lead to the implication that infinity is a real number but I am not willing to concede that these rules are part of the definition of a real number. For that, you would need to present some third party source which say the rules you posed are part of the definition of real numbers rather than merely some properties which can be derived by only considering real numbers in the neighborhood of the origin. The wikipedia page says that \((\mathbb{R},+,\cdot)\) is a field, in particular \((\mathbb{R},+)\) is a group, which implies the three properties that I mentioned, if you add a simple notational convention. I do not know if you will consider the defining properties of a group as only "derived". 2 hours ago, sevensixtwo said: Until you do, I will stick with the definition in my real analysis book. If you have such a book it is probably in yours too: real numbers are cuts in the real number line. No, I have to admit that I never came across such a definition. What is the name of your book? Maybe I will recollect. 2 hours ago, sevensixtwo said: I just tried to google it and Wolfram doesn't define real numbers but I found the same definition on Wikipedia. I can post the link to Wikipedia if you want. You mean the page on "Real Number"? I would like to know where on that page you can find this? I see mention of "Dedekind cuts", but I would be concerned to define real numbers in that way, it is only one of many possible constructions of reals, and the other ones are just as good. It is not a definition anyway, only a construction. Edited October 5 by taeto 0 Share this post Link to post Share on other sites

sevensixtwo 18 Posted October 5 Pretty much, yup. 0 Share this post Link to post Share on other sites

taeto 37 Posted October 5 2 hours ago, sevensixtwo said: Since you seem to have a brain and are not trolling me like user name "stupid idiot," what is your opinion on an implication that infinity is a real number? Do not worry. Nobody in this forum will troll you, the moderators are too strict. You seem confident when you use the word "infinity" that it means something specific to you. To me it means nothing in particular. For someone who works with elliptic curves, infinity is just an extra point that gets added to get the full group structure. It could be that infinity is something like the set with one element \(\{0\}\). Of course it is the name of the symbol \(\infty\) that gets used all the time in indefinite summations and integrals, and in limit calculations. It is just a letter in that context. So if you say "let infinity be a real number", then I am cool with that, except I would find "let x be a real number" easier to live with. But in contrast you mean something precise and absolute when you say "infinity"? 0 Share this post Link to post Share on other sites

wtf 95 Posted October 5 (edited) 3 hours ago, sevensixtwo said: real numbers are cuts in the real number line. But this couldn't be. You are defining the real numbers as cuts in ... what? In the real number line. Where do those real numbers come form before you have defined them? Your definition is circular. Of course the answer is that the real numbers are cuts in the rationals. You should demand your money back from the university that sold you a course in real analysis, since you clearly didn't learn anything. Edited October 5 by wtf 1 Share this post Link to post Share on other sites

studiot 1471 Posted October 5 9 minutes ago, wtf said: But this couldn't be. You are defining the real numbers as cuts in ... what? In the real number line. Where do those real numbers come form before you have defined them? Your definition is circular. Of course the answer is that the real numbers are cuts in the rationals. You should demand your money back from the university that sold you a course in real analysis, since you clearly didn't learn anything. +1 0 Share this post Link to post Share on other sites

taeto 37 Posted October 5 1 hour ago, taeto said: You mean the page on "Real Number"? I would like to know where on that page you can find this? I see mention of "Dedekind cuts", but I would be concerned to define real numbers in that way, it is only one of many possible constructions of reals, and the other ones are just as good. It is not a definition anyway, only a construction. I decide to take this seriously, because I just see it over and over, with lots of misunderstanding mixed in. Take another example than the concept of "real numbers". Let us say we talk instead about "cars". We look up the wikipedia definition "A car (or automobile) is a wheeled motor vehicle used for transportation. It runs primarily on roads, seats one to eight people, has four tires, and mainly transports people rather than goods." You may agree that by this description you now have a firm basis for recognizing when some object in your vicinity is a car or not. Now someone says that Porsche has provided a construction of a car, so thereby and henceforth, the definition of a car has to be something that is made by the Porsche factories. The big news, apparently, on this forum is that: no, that is not how definitions work. In the case of real numbers, or in indeed all mathematical objects, the first thing that comes about is a description of what are the supposed properties of such a thing. In the case of real numbers it happens to be not the number of wheels etc., but things like properties of addition and multiplication, ordering properties, and more. When you have those properties sorted out, you are ready to distinguish between objects that are real numbers and those that are not. If somebody says, wait, Cauchy has a construction of real numbers, so we should take the definition of real numbers to be equivalence classes of Cauchy sequences, then that is BS, it is still not how it works. Not with cars, not here either. 0 Share this post Link to post Share on other sites

uncool 202 Posted October 5 10 hours ago, sevensixtwo said: (1) Yes I can. You can make your same argument about "cannot" regarding the square root of a negative number and yet we use those every day. (2) I use exactly the usual definition of a neighborhood of infinity. You are totally wrong. If there is some inconsistency with the definition of a real number number as a cut in the real number line then I challenge you to identify it. If there is some other inconsistency besides that, then those are only properties of real numbers in the neighborhood of the origin. (3) I am discussing a region of the complex plane which has been previously neglected but the neighborhood of infinity is absolutely in the complex plane. Again, you are totally wrong. 1) Using the standard definition of real numbers, you can't. If you think you have a definition or construction that will allow it, you will have to provide it; in doing so, you may have to demonstrate the properties you usually take for granted. 2) You defined (or claimed) "neighborhood of infinity" to mean numbers that can be expressed as "infinity hat minus a real number". That is not standard at all, and would break the usual real numbers. 3) You are discussing your own construction, which is not the usual complex numbers. No complex number (or real number) is of the form "infinity hat minus b". 3 Share this post Link to post Share on other sites

sevensixtwo 18 Posted October 6 Nah, you are wrong. -4 Share this post Link to post Share on other sites

studiot 1471 Posted October 6 24 minutes ago, sevensixtwo said: Nah, you are wrong. Then you will easily be able to demonstrate the falshood of uncool's statements. Assertion alone is not proof. +1 to uncool 0 Share this post Link to post Share on other sites

uncool 202 Posted October 6 (edited) I believe you've said you favor the "Dedekind cut". There are two ways the Dedekind cut can be used to define the real numbers: as part of a definition, and as part of a construction. Definition: The real numbers are an ordered field so that for every Dedekind cut (depending on the definition used, you may need to add "nontrivial"), either the lower or upper set has a maximum or minimum, respectively. If you are using this, then the set of real numbers of the form "infinity hat minus b" are larger than anything not of that form, so the two (that set and its complement) form a Dedekind cut. What is the corresponding maximum or minimum? Construction: The real numbers are the set of Dedekind cuts of rational numbers. If younare using this, then please describe the Dedekind cut of rational numbers that gives the "real number" "infinity hat minus 5". Edited October 6 by uncool 1 Share this post Link to post Share on other sites

taeto 37 Posted October 7 On 10/6/2018 at 9:43 AM, sevensixtwo said: Nah, you are wrong. Three questions: 1. Who is wrong? 2. What are they wrong about? 3. What would be right? 1 Share this post Link to post Share on other sites

studiot 1471 Posted October 7 8 minutes ago, taeto said: Three questions: 1. Who is wrong? 2. What are they wrong about? 3. What would be right? +1 0 Share this post Link to post Share on other sites

wtf 95 Posted October 8 (edited) I'm only following sporadically but this is what I'm confused about from the beginning. * If [math]\widehat{0} = 0[/math] as has been recently acknowledged by OP; and * If correspondingly, [math]\widehat{\infty} = \infty[/math]; and * If the symbol [math]\infty[/math] is identical to [math]+ \infty[/math] in the extended real numbers; which I believe the OP acknowledged a while back; * Then the expression [math]\widehat{\infty} - b[/math] is undefined. It can't be sensibly defined. The exposition so far has not defined it. Has the OP addressed this issue yet? Also, I'm still confused about the neighborhood of infinity. In the real or extended real numbers, we can certainly call the set of reals larger than some fixed real, a neighborhood of infinity. And if we consider the set of reals whose absolute value is greater than some fixed real,then that would be a neighborhood of infinity in the space where you identify plus and minus infinity of the extended reals. In other words it's a circle, but with the point diametrically opposed 0 called infinity. It's a regular circle but with a funny metric. The two-dimensional analog of that construction shows how to regard the complex numbers as a sphere. If you take the complex plane you can add a single point called [math]\infty[/math], which can be visualized as the north pole of a sphere. This is called the Riemann sphere, https://en.wikipedia.org/wiki/Riemann_sphere It's like going "out to infinity" in every direction and sewing together the boundary at infinity to make a sphere. Since there's no order on the complex numbers, there's only one point at infinity. Another name for this construction is the one-point compactification of the plane. If you know what a compact set is, you know the plane's not compact. But using this construction you can add a single point and make it a compact set. If you draw a circle on the sphere centered at the north pole, you could legitimately call that a neighborhood of infinity of the complex numbers. It's the same as if you consider the set of all complex numbers whose modulus, or absolute value, is greater than some fixed nonnegative real number. Now as I understand it, the OP does not mean any of those things. I still don't understand what "the" neighborhood of infinity is; and I also don't understand why OP calls it "the" instead of "a", since in the contexts I mentioned, there are many neighborhoods of infinity. Edited October 8 by wtf 0 Share this post Link to post Share on other sites

hypervalent_iodine 1216 Posted October 12 ! Moderator Note It is clear the OP has no interest in discussing this further. As such, thread closed. 0 Share this post Link to post Share on other sites